When a nuisance parameter is unidentified under the null hypothesis, standard testing procedures cannot be applied due to the singularity of the information matrix. Probably best known examples are the problems of unknown change points and the mixtures of distributions in econometrics and statistics. Davies (1977, 1987) proposes a general solution to this type of problems.In this dissertation, we study three applications: tests for parameter constancy, white noise against the autoregressive moving average (ARMA(1,1)) alternative, and autoregressive conditional heteroskedasticity in mean (ARCH-M) model. Davies' procedure and the conventional Lagrange multiplier (LM) test are applied, and find that Davies' test outperforms the LM test. However, despite of its generality, Davies' approach has several deficiencies to be implemented for more general cases and it is quite expensive computationally.For testing for parameter constancy, a joint LM test for autocorrelation and heteroskedasticity is suggested as a simple alternative test to Davies' procedure. For testing white noise against ARMA(1,1), we implement a more exact and simplified version of Davies approach. Monte Carlo results indicate that both the joint LM test and the simplified version of Davies have good finite sample power properties.